Further, let \((\Omega ,\mathcal {F},\mathbb {P}_{z},\lbrace Z_n\rbrace _{n\ge 0},\lbrace \mathcal {F}_n\rbrace _{n\ge 0})_{z\in \mathsf {Z}}\) , denoted by \(\lbrace Z_n\rbrace _{n\ge 0}\)  in the sequel, be a temporally-homogeneous Markov chain on \((\mathsf {Z},\mathfrak {B}(\mathsf {Z}))\) , in the sense of [1]}.
The process \(\lbrace Z_n\rbrace _{n\ge 0}=\lbrace (\mathrm {pr}_\mathsf {X}(Z_n),\mathrm {pr}_\mathsf {Y}(Z_n))\rbrace _{n\ge 0}\)  is viewed as a sequence of input data (the first component) together with their outputs (the second component).
